Fractals and Chaos: An Illustrated Course

Beschreibung

Beschreibung

Fractals and Chaos: An Illustrated Course provides you with a practical, elementary introduction to fractal geometry and chaotic dynamics-subjects that have attracted immense interest throughout the scientific and engineering disciplines. The book may be used in part or as a whole to form an introductory course in either or both subject areas. A prominent feature of the book is the use of many illustrations to convey the concepts required for comprehension of the subject. In addition, plenty of problems are provided to test understanding. Advanced mathematics is avoided in order to provide a concise treatment and speed the reader through the subject areas. The book can be used as a text for undergraduate courses or for self-study.

Inhaltsverzeichnis

INTRODUCTION Introduction A matter of fractals Deterministic chaos Chapter summary and further reading REGULAR FRACTALS AND SELF-SIMILARITY Introduction The Cantor set Non-fractal dimensions: the Euclidean and topological dimension The similarity dimension The Koch curve The quadratic Koch curve The Koch island Curves in the plane with similarity dimension exceeding 2 The Sierpinski gasket and carpet The Menger Sponge Chapter summary and further reading Revision questions and further tasks RANDOM FRACTALS Introduction Randomizing the Cantor set and Koch curve Fractal boundaries The box counting dimension and the Hausdorff dimension The structured walk technique and the divider dimension The perimeter-area relationship Chapter summary and further reading Revision questions and further tasks REGULAR AND FRACTIONAL BROWNIAN MOTION Introduction Regular Brownian motion Fractional Brownian motion: time traces Fractional Brownian surfaces Fractional Brownian motion: spatial trajectories Diffusion limited aggregation The color and power and noise Chapter summary and further reading Revision questions and further tasks ITERATIVE FEEDBACK PROCESSES AND CHAOS Introduction Population growth and the Verhulst model The logistic map The effect of variation in the control parameter General form of the iterated solutions of the logistic map Graphical iteration of the logistic map Bifurcation, stability and the Feigenbaum number A two dimensional map: the Henon model Iterations in the complex plane: Julia sets and the Mandelbrot set Chapter summary and further reading Revision questions and further tasks CHAOTIC OSCILLATIONS Introduction A simple nonlinear mechanical oscillator: the Duffing oscillator Chaos in the weather: the Lorenz model The Rossler systems Phase space, dimension and attractor form Spatially extended systems: coupled oscillators Spatially extended systems: fluids Mathematical routes to chaos and turbulence Chapter summary and further reading Revision questions and further tasks CHARACTERIZING CHAOS Introduction Preliminary characterization: visual inspection Preliminary characterization: frequency spectra Characterizing chaos: Lyapunov exponents Characterizing chaos: dimension estimates Attractor reconstruction The embedding dimension for attractor reconstruction The effect of noise Regions of behavior on the attractor and characterization limitations Chapter summary and further reading Revision questions and further task APPENDIX 1: Computer Program for Lorenz Equations APPENDIX 2: Illustrative Papers APPENDIX 3: Experimental Chaos SOLUTIONS REFERENCES